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(http://www.gaf.ni.ac.yu/CDP/Assignment-IX.pdf) A course in Numerical Methods in Computational Engineering oriented to engineering.
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Study Guide Bachelor and Master in Computational Engineering
Computational Engineering. Academic Year 2015/2016. August 5 2015. Vorbemerkung. Grundlage für dieses Dokument ist die Fachprüfungsordnung des Bachelor-
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Computational Science and Engineering M
The goal of this track is to produce scientists and engineers with focus training and application in computational sciences
Computational Engineering — Introduction to Numerical Methods
students of engineering disciplines but also for computational engineers in industrial practice. Many of the methods presented are integrated in the flow.
UNIVERSITY of NIS GAF
FACULTY OF CIVIL ENGINEERING AND ARCHITECTURE
NUMERICAL METHODS
In Computational Engineering
financed by ·Austrian
G.V. MILOVANOVIC, sci. advisor
D. R. DORDEVIC, lecturer
Development Cooperation
Nis, 2001
UNIVERSITY of NIS
FACULTY OF CIVIL ENGINEERING AND ARCHITECTURE
NUMERICAL METHODS
in Computational EngineeringG.V. MILOVANOVIC, sci. advisor
(http:/ /gauss.elfak.ni.ac.yu/)D. R. DORDEVIC, lecturer
(http:/ jwww.gaf.ni.ac.yu/cdp/lecturer.htm)Nis, 2007.
Prof. Gradimir V. ?IIilm·;moYit. lTni\'l'rsitY
Prof. Don1e R. Don1PYi(, Unin'rsitY ·
Numerical Methods in Computational Engineering
ISBN 978 86-80295-81-7
The publishing of this script is part of the project CDP+ 53/2006 financed by AustrianCooperation through WUS Austria
This copy is not for sale
Ova skripta je objavljena u okviru projekta WUS Austria CDP+ 53/2006 finansiranog od strane Austrian CooperationBesplatan primerak
Printed hy Grafilm Petkoyi(. Nis, 2007 (100 copies)Contents
Preface .
1. Mathematics and Computer Science
1.1. Calculus
1.2. Number represeut.at.ion
1.3. Error, accuracy, aud stability
1.4. Programming
1.5. Numerical :-;oftware
1.6. Case stu Bibliography .
I (http://www.gaf.ni.ac.yu/CDP/Assignment-I.pdf)
IX 1 l 10 10 11 12 2. Linear Systems of Algebraic Equations: Direct Methods 15
2 .1. Elements of matrix cakulu:-; . l;)
2.1.1. LR. factmization of quadratic matrix lG
2.1.2. l'viat.rix cige!lvPct.nrs awl eigenvalues 17
2.2. Direct met.hocl:-; in linear algebra 17
2.2.1.
Iut.ro 2.2.2.
Gauss with pivoting 18
2.2.3. lVIatrix
iHver:-;iou usiug Gauss method 21 2.2.4. Factorization md.hods 21
2.2.5. Program realization 24
Bibliography .
3. Linear Systems of Algebraic Equations: Iterative Methods :3;J
3.1. Intmcluc:tiou :35
3.2. Simple iteration metho 3.3. Gauss-Seidel method . 3G
3.4. Program 37
3.5. Packages fm systems of linear algebraic equation:-; 41
Bibliography . 42
Assignment-II-III (http://www. ni. ac. yu/CDP I Assignment-II-III. pdf) 4. Problems 45
4.1. Introduction 4ij
4.2. Localization of eigenvnlues 50
4.3. l'viethocls for dominant eigenvalnes 51
4.4. Niethocls for subdominant G3
4.5, problem for synnnetric tridiagonal matrices 57
4,6. LR ancl QR algorithms . :)9
4.7. Soft1va.re eigenpacka.ges GO
4.8. Generalized a.nd nonlinear eigenvalue problems G1
v 62
BibliogTaphy .
Assignment-
IV 5. Nonlinear Equations and Systems of Equations
5 .1. N onlineai' equations
5.1.0. Introduction
5.1.1. Newton's method
5.1.2. Bisection method
5 .1.3. Program reali,.-;ation
65
65
65
68
72
73
5.2. System of nonlinear equations 83
5.2.1. Newton-Kantorowitch (R.aphson) method 83
5.2.2. Gradient method 87
5.2.3. Globally
convergent methods 91 Bibliography . 94
Assignment-V (http;/ lwww. gaf. ni. ac. yuiCDP I Assignment-V. pdf) 6. Approximation and Interpolation 97
6.1. Introduction 97
6.2. Chebyshev systems
6.3. Lagrange's interpolation
6.4. Newton's interpolation with divided differences
6.5. Newton's
interpolation formulas 6.6. Spline functions and interpolation by splines
6.7. Prony's interpolation
6.8. Packages for interpolation of functions
Bibliography .
Assignment-VI (http: I lwww. gaf .ni. ac. yuiCDP I Assignment-VI. pdf) 98
99
100
102
104
106
107
108
7. Best Approximation of Functions 109
7.1. Introduction 109
7.2. Best L
2 approxiination 7.3. Best I"· approximation
7.4. Packages for approximation of fnnctions
Bibliography .
Assignment-VII . (http: I lwww. gaf. ni. ac. yuiCDPI Assignment-VII. pdf) 112
114
119
119
8. Numerical Differentiation and Integration 121
8.1. Numerical differentiation 121
8.1.1.
Introduction 121
8.1.2. Formulas for numerical differentiation 121
8.2. Numerical
integration-Quadrature formulas 12.3 8.2.1. Introduction 123
8.2.2.
Newton-Cotes formulas 124
8.2.3.
Generalized qHadra.tHne formulas 126
8.2.4.
integration 128 8.2.5.
Program 128
8.2.6.
On numerical evaluation of a class of double integrals 132 8.2.7.
Packages for nmnerical integration 134
Bil>liography . 13.5
Assignnwnt-VIII (http: I lwww. gaf. ni. ac. yuiCDP I Assignment-VIII. pdf) 9. Ordinary Differential Equations -ODE 137
9.1. Introduction 1:3 7
vi 0.2. Euler's method
0.3. Ge1wr;d liiwm multi-st.<p uwthod
0.4. of initial valuPs
0.:). Predict.or-corrP.ct.or llH'.tlwcls
0.6. Program n·alizat.ion of methods
0.7. RungP-Kut.t.a lll<-'t.lwds
0.8. Progrmu realization of Rnng<'-Kut.ta methods
0.0. Solution of of <-'quat.ious and equations of higher order
0.10. Bmmdary prohbus
0.11. Packag Bibliography .
Assignment-IX
10. Partial Differential Equations -PDE
10.1. Introduction
10.2. Grid method .
10.3. Laplace eqnation
10.4. Vlave eqnat.ion
10.5. Packages
for Bibliography
Assignment-X
11. Integral Equations .
11.1. Introclnction
11.2. Mc!t.lwcl of approximations
11.3. Application of q1wdra.t.me formnlas
1.38 1:30 141
1-.11 142
14.) 1;j() l;j:j 158
Hil 161
1Ci3 163
164
165
167
160
170
173
173
17:) 175
176 11.4. Program
Bibliography
Assignnwnt-XI
178
Appendices
A.l. Equations of Technical Physics
A.2. Special Functions
A.3. Numerical Methods in FEM
A.4. Numerical Methods in Informatics
vii (http://www.gaf.ni.ac.yu/CDP/ETPH.pdf) (http://www.gaf.ni.ac.yu/CDP/SPEC.pdf) (http://www.gaf.ni.ac.yu/CDP/NMFEM.pdf) (http://www.gaf.ni.ac.yu/CDP/NMINF.pdf) Preface
A course in Numerical Methods in Computational Engineering, oriented to engineering education, originates at first from the course in numerical analysis for graduate students of Faculty of Civil Engineering and Architecture of Nis (GAF), and then from course Numer ical Methods held in English language at Faculty of Civil Engineering in Belgrade in the frame of project DYNET (Dynamical Network) in common of Faculty of Civil Engineering of University of Bochum, Faculty of Civil Engineering and Architecture of University of Nis, Faculty of Civil Engineering of University Belgrade, and IZIIS (Institute for Earth quake Engineering and Seismology) of University Skopje. The subject Numerical Analysis was held in the first semester of postgraduate studies at GAF by Prof G. V. Milovanovic for years. In continuation, following Bologna process, the new structured subject entitled Numerical Analysis is be introduced to PhD students at GAF. In addition, having in n1ind that course in numerical analysis become accepted as an important ingredient in the undergraduate education in engineering and technology, it was with its main topics involved in undergraduate subject Informatics II at GAF Nis (As a collateral case, in Appendix A.4. -in electronic fonn -are given numerical methods in Informatics, what could be interesting for students of this orientation). The backbone of this script are famous books of G. V. Milovanovic, Numerical Anal ysis, Part I, II, and III, Naucna knjiga, Beograd, 1988 (Serbian). In addition, the book Programming Numerical Methods in Fortran, by G. V. Milovanovic and Dj. R. Djordjevic, University of Nis, 1981 (Serbian), with its engineering-oriented text and codes, was rather used. As previously noted, this textbook is supporting undergraduate studies, master and doctoral study at GAF; and international master study in the frame of DYNET project. Presentation on GAF site would enable distance technique and on-line consulta tions with lecturer. By up-to-day engineering oriented applications the supporting of life long education of civil engineers will be enabled. This script will be available on the site of GAF (http: I lwww. gaf. ni. ac. yu) under In ternational Projects and can be reached by chapters using address http: I lwww. gaf. ni. ac. yul cdplsubj ect_syllabus .htm. Each chapter concludes with a ba sic bibliography and suggested further reading. Tutorial exercises in form of selected as signments are also presented on the site of GAF. Some hints for solutions are given in the same files. Devoted primarily to students of Civil Engineering (undergraduate and graduate - master & PhD), this textbook is dedicated also to industry and research purposes. Authors
lX Faculty of Civil
Belgrade
Master Study
Faculty of Civil Engineering and Architectme
Nis Doctoral Study
COMPUTATIONAL ENGINEERING
LECTURES
LESSON I
1. Mathematics and Computer Science
1.1 Calculus
The principal topics in calculus are the real and complex number systems, the concept of limits ancl convergence, and the properties of functions. Convergence of a sequence of numbers :ri is defined as follows: The sequence :r.;, converges to the limit :c* if
7 given any tolerance E > 0, there is an index N = N(E) so that for all ·i 2:: N we have I xi-:r* I ::=;E. The notation for this is lim x.;, = x*. i--too Convergence is also a principal topics of numerical computation, but with a different emphasis.quotesdbs_dbs17.pdfusesText_23
Bibliography .
I (http://www.gaf.ni.ac.yu/CDP/Assignment-I.pdf)
IX 1 l 10 10 11 122. Linear Systems of Algebraic Equations: Direct Methods 15
2 .1. Elements of matrix cakulu:-; . l;)
2.1.1. LR. factmization of quadratic matrix lG
2.1.2. l'viat.rix cige!lvPct.nrs awl eigenvalues 17
2.2.Direct met.hocl:-; in linear algebra 17
2.2.1.
Iut.ro 2.2.2.
Gauss with pivoting 18
2.2.3. lVIatrix
iHver:-;iou usiug Gauss method 21 2.2.4. Factorization md.hods 21
2.2.5. Program realization 24
Bibliography .
3. Linear Systems of Algebraic Equations: Iterative Methods :3;J
3.1. Intmcluc:tiou :35
3.2. Simple iteration metho 3.3. Gauss-Seidel method . 3G
3.4. Program 37
3.5. Packages fm systems of linear algebraic equation:-; 41
Bibliography . 42
Assignment-II-III (http://www. ni. ac. yu/CDP I Assignment-II-III. pdf) 4. Problems 45
4.1. Introduction 4ij
4.2. Localization of eigenvnlues 50
4.3. l'viethocls for dominant eigenvalnes 51
4.4. Niethocls for subdominant G3
4.5, problem for synnnetric tridiagonal matrices 57
4,6. LR ancl QR algorithms . :)9
4.7. Soft1va.re eigenpacka.ges GO
4.8. Generalized a.nd nonlinear eigenvalue problems G1
v 62
BibliogTaphy .
Assignment-
IV 5. Nonlinear Equations and Systems of Equations
5 .1. N onlineai' equations
5.1.0. Introduction
5.1.1. Newton's method
5.1.2. Bisection method
5 .1.3. Program reali,.-;ation
65
65
65
68
72
73
5.2. System of nonlinear equations 83
5.2.1. Newton-Kantorowitch (R.aphson) method 83
5.2.2. Gradient method 87
5.2.3. Globally
convergent methods 91 Bibliography . 94
Assignment-V (http;/ lwww. gaf. ni. ac. yuiCDP I Assignment-V. pdf) 6. Approximation and Interpolation 97
6.1. Introduction 97
6.2. Chebyshev systems
6.3. Lagrange's interpolation
6.4. Newton's interpolation with divided differences
6.5. Newton's
interpolation formulas 6.6. Spline functions and interpolation by splines
6.7. Prony's interpolation
6.8. Packages for interpolation of functions
Bibliography .
Assignment-VI (http: I lwww. gaf .ni. ac. yuiCDP I Assignment-VI. pdf) 98
99
100
102
104
106
107
108
7. Best Approximation of Functions 109
7.1. Introduction 109
7.2. Best L
2 approxiination 7.3. Best I"· approximation
7.4. Packages for approximation of fnnctions
Bibliography .
Assignment-VII . (http: I lwww. gaf. ni. ac. yuiCDPI Assignment-VII. pdf) 112
114
119
119
8. Numerical Differentiation and Integration 121
8.1. Numerical differentiation 121
8.1.1.
Introduction 121
8.1.2. Formulas for numerical differentiation 121
8.2. Numerical
integration-Quadrature formulas 12.3 8.2.1. Introduction 123
8.2.2.
Newton-Cotes formulas 124
8.2.3.
Generalized qHadra.tHne formulas 126
8.2.4.
integration 128 8.2.5.
Program 128
8.2.6.
On numerical evaluation of a class of double integrals 132 8.2.7.
Packages for nmnerical integration 134
Bil>liography . 13.5
Assignnwnt-VIII (http: I lwww. gaf. ni. ac. yuiCDP I Assignment-VIII. pdf) 9. Ordinary Differential Equations -ODE 137
9.1. Introduction 1:3 7
vi 0.2. Euler's method
0.3. Ge1wr;d liiwm multi-st.<p uwthod
0.4. of initial valuPs
0.:). Predict.or-corrP.ct.or llH'.tlwcls
0.6. Program n·alizat.ion of methods
0.7. RungP-Kut.t.a lll<-'t.lwds
0.8. Progrmu realization of Rnng<'-Kut.ta methods
0.0. Solution of of <-'quat.ious and equations of higher order
0.10. Bmmdary prohbus
0.11. Packag Bibliography .
Assignment-IX
10. Partial Differential Equations -PDE
10.1. Introduction
10.2. Grid method .
10.3. Laplace eqnation
10.4. Vlave eqnat.ion
10.5. Packages
for Bibliography
Assignment-X
11. Integral Equations .
11.1. Introclnction
11.2. Mc!t.lwcl of approximations
11.3. Application of q1wdra.t.me formnlas
1.38 1:30 141
1-.11 142
14.) 1;j() l;j:j 158
Hil 161
1Ci3 163
164
165
167
160
170
173
173
17:) 175
176 11.4. Program
Bibliography
Assignnwnt-XI
178
Appendices
A.l. Equations of Technical Physics
A.2. Special Functions
A.3. Numerical Methods in FEM
A.4. Numerical Methods in Informatics
vii (http://www.gaf.ni.ac.yu/CDP/ETPH.pdf) (http://www.gaf.ni.ac.yu/CDP/SPEC.pdf) (http://www.gaf.ni.ac.yu/CDP/NMFEM.pdf) (http://www.gaf.ni.ac.yu/CDP/NMINF.pdf) Preface
A course in Numerical Methods in Computational Engineering, oriented to engineering education, originates at first from the course in numerical analysis for graduate students of Faculty of Civil Engineering and Architecture of Nis (GAF), and then from course Numer ical Methods held in English language at Faculty of Civil Engineering in Belgrade in the frame of project DYNET (Dynamical Network) in common of Faculty of Civil Engineering of University of Bochum, Faculty of Civil Engineering and Architecture of University of Nis, Faculty of Civil Engineering of University Belgrade, and IZIIS (Institute for Earth quake Engineering and Seismology) of University Skopje. The subject Numerical Analysis was held in the first semester of postgraduate studies at GAF by Prof G. V. Milovanovic for years. In continuation, following Bologna process, the new structured subject entitled Numerical Analysis is be introduced to PhD students at GAF. In addition, having in n1ind that course in numerical analysis become accepted as an important ingredient in the undergraduate education in engineering and technology, it was with its main topics involved in undergraduate subject Informatics II at GAF Nis (As a collateral case, in Appendix A.4. -in electronic fonn -are given numerical methods in Informatics, what could be interesting for students of this orientation). The backbone of this script are famous books of G. V. Milovanovic, Numerical Anal ysis, Part I, II, and III, Naucna knjiga, Beograd, 1988 (Serbian). In addition, the book Programming Numerical Methods in Fortran, by G. V. Milovanovic and Dj. R. Djordjevic, University of Nis, 1981 (Serbian), with its engineering-oriented text and codes, was rather used. As previously noted, this textbook is supporting undergraduate studies, master and doctoral study at GAF; and international master study in the frame of DYNET project. Presentation on GAF site would enable distance technique and on-line consulta tions with lecturer. By up-to-day engineering oriented applications the supporting of life long education of civil engineers will be enabled. This script will be available on the site of GAF (http: I lwww. gaf. ni. ac. yu) under In ternational Projects and can be reached by chapters using address http: I lwww. gaf. ni. ac. yul cdplsubj ect_syllabus .htm. Each chapter concludes with a ba sic bibliography and suggested further reading. Tutorial exercises in form of selected as signments are also presented on the site of GAF. Some hints for solutions are given in the same files. Devoted primarily to students of Civil Engineering (undergraduate and graduate - master & PhD), this textbook is dedicated also to industry and research purposes. Authors
lX Faculty of Civil
Belgrade
Master Study
Faculty of Civil Engineering and Architectme
Nis Doctoral Study
COMPUTATIONAL ENGINEERING
LECTURES
LESSON I
1. Mathematics and Computer Science
1.1 Calculus
The principal topics in calculus are the real and complex number systems, the concept of limits ancl convergence, and the properties of functions. Convergence of a sequence of numbers :ri is defined as follows: The sequence :r.;, converges to the limit :c* if
7 given any tolerance E > 0, there is an index N = N(E) so that for all ·i 2:: N we have I xi-:r* I ::=;E. The notation for this is lim x.;, = x*. i--too Convergence is also a principal topics of numerical computation, but with a different emphasis.quotesdbs_dbs17.pdfusesText_23
2.2.2.
Gauss with pivoting 18
2.2.3. lVIatrix
iHver:-;iou usiug Gauss method 212.2.4. Factorization md.hods 21
2.2.5. Program realization 24
Bibliography .
3. Linear Systems of Algebraic Equations: Iterative Methods :3;J
3.1. Intmcluc:tiou :35
3.2. Simple iteration metho 3.3. Gauss-Seidel method . 3G
3.4. Program 37
3.5. Packages fm systems of linear algebraic equation:-; 41
Bibliography . 42
Assignment-II-III (http://www. ni. ac. yu/CDP I Assignment-II-III. pdf) 4. Problems 45
4.1. Introduction 4ij
4.2. Localization of eigenvnlues 50
4.3. l'viethocls for dominant eigenvalnes 51
4.4. Niethocls for subdominant G3
4.5, problem for synnnetric tridiagonal matrices 57
4,6. LR ancl QR algorithms . :)9
4.7. Soft1va.re eigenpacka.ges GO
4.8. Generalized a.nd nonlinear eigenvalue problems G1
v 62
BibliogTaphy .
Assignment-
IV 5. Nonlinear Equations and Systems of Equations
5 .1. N onlineai' equations
5.1.0. Introduction
5.1.1. Newton's method
5.1.2. Bisection method
5 .1.3. Program reali,.-;ation
65
65
65
68
72
73
5.2. System of nonlinear equations 83
5.2.1. Newton-Kantorowitch (R.aphson) method 83
5.2.2. Gradient method 87
5.2.3. Globally
convergent methods 91 Bibliography . 94
Assignment-V (http;/ lwww. gaf. ni. ac. yuiCDP I Assignment-V. pdf) 6. Approximation and Interpolation 97
6.1. Introduction 97
6.2. Chebyshev systems
6.3. Lagrange's interpolation
6.4. Newton's interpolation with divided differences
6.5. Newton's
interpolation formulas 6.6. Spline functions and interpolation by splines
6.7. Prony's interpolation
6.8. Packages for interpolation of functions
Bibliography .
Assignment-VI (http: I lwww. gaf .ni. ac. yuiCDP I Assignment-VI. pdf) 98
99
100
102
104
106
107
108
7. Best Approximation of Functions 109
7.1. Introduction 109
7.2. Best L
2 approxiination 7.3. Best I"· approximation
7.4. Packages for approximation of fnnctions
Bibliography .
Assignment-VII . (http: I lwww. gaf. ni. ac. yuiCDPI Assignment-VII. pdf) 112
114
119
119
8. Numerical Differentiation and Integration 121
8.1. Numerical differentiation 121
8.1.1.
Introduction 121
8.1.2. Formulas for numerical differentiation 121
8.2. Numerical
integration-Quadrature formulas 12.3 8.2.1. Introduction 123
8.2.2.
Newton-Cotes formulas 124
8.2.3.
Generalized qHadra.tHne formulas 126
8.2.4.
integration 128 8.2.5.
Program 128
8.2.6.
On numerical evaluation of a class of double integrals 132 8.2.7.
Packages for nmnerical integration 134
Bil>liography . 13.5
Assignnwnt-VIII (http: I lwww. gaf. ni. ac. yuiCDP I Assignment-VIII. pdf) 9. Ordinary Differential Equations -ODE 137
9.1. Introduction 1:3 7
vi 0.2. Euler's method
0.3. Ge1wr;d liiwm multi-st.<p uwthod
0.4. of initial valuPs
0.:). Predict.or-corrP.ct.or llH'.tlwcls
0.6. Program n·alizat.ion of methods
0.7. RungP-Kut.t.a lll<-'t.lwds
0.8. Progrmu realization of Rnng<'-Kut.ta methods
0.0. Solution of of <-'quat.ious and equations of higher order
0.10. Bmmdary prohbus
0.11. Packag Bibliography .
Assignment-IX
10. Partial Differential Equations -PDE
10.1. Introduction
10.2. Grid method .
10.3. Laplace eqnation
10.4. Vlave eqnat.ion
10.5. Packages
for Bibliography
Assignment-X
11. Integral Equations .
11.1. Introclnction
11.2. Mc!t.lwcl of approximations
11.3. Application of q1wdra.t.me formnlas
1.38 1:30 141
1-.11 142
14.) 1;j() l;j:j 158
Hil 161
1Ci3 163
164
165
167
160
170
173
173
17:) 175
176 11.4. Program
Bibliography
Assignnwnt-XI
178
Appendices
A.l. Equations of Technical Physics
A.2. Special Functions
A.3. Numerical Methods in FEM
A.4. Numerical Methods in Informatics
vii (http://www.gaf.ni.ac.yu/CDP/ETPH.pdf) (http://www.gaf.ni.ac.yu/CDP/SPEC.pdf) (http://www.gaf.ni.ac.yu/CDP/NMFEM.pdf) (http://www.gaf.ni.ac.yu/CDP/NMINF.pdf) Preface
A course in Numerical Methods in Computational Engineering, oriented to engineering education, originates at first from the course in numerical analysis for graduate students of Faculty of Civil Engineering and Architecture of Nis (GAF), and then from course Numer ical Methods held in English language at Faculty of Civil Engineering in Belgrade in the frame of project DYNET (Dynamical Network) in common of Faculty of Civil Engineering of University of Bochum, Faculty of Civil Engineering and Architecture of University of Nis, Faculty of Civil Engineering of University Belgrade, and IZIIS (Institute for Earth quake Engineering and Seismology) of University Skopje. The subject Numerical Analysis was held in the first semester of postgraduate studies at GAF by Prof G. V. Milovanovic for years. In continuation, following Bologna process, the new structured subject entitled Numerical Analysis is be introduced to PhD students at GAF. In addition, having in n1ind that course in numerical analysis become accepted as an important ingredient in the undergraduate education in engineering and technology, it was with its main topics involved in undergraduate subject Informatics II at GAF Nis (As a collateral case, in Appendix A.4. -in electronic fonn -are given numerical methods in Informatics, what could be interesting for students of this orientation). The backbone of this script are famous books of G. V. Milovanovic, Numerical Anal ysis, Part I, II, and III, Naucna knjiga, Beograd, 1988 (Serbian). In addition, the book Programming Numerical Methods in Fortran, by G. V. Milovanovic and Dj. R. Djordjevic, University of Nis, 1981 (Serbian), with its engineering-oriented text and codes, was rather used. As previously noted, this textbook is supporting undergraduate studies, master and doctoral study at GAF; and international master study in the frame of DYNET project. Presentation on GAF site would enable distance technique and on-line consulta tions with lecturer. By up-to-day engineering oriented applications the supporting of life long education of civil engineers will be enabled. This script will be available on the site of GAF (http: I lwww. gaf. ni. ac. yu) under In ternational Projects and can be reached by chapters using address http: I lwww. gaf. ni. ac. yul cdplsubj ect_syllabus .htm. Each chapter concludes with a ba sic bibliography and suggested further reading. Tutorial exercises in form of selected as signments are also presented on the site of GAF. Some hints for solutions are given in the same files. Devoted primarily to students of Civil Engineering (undergraduate and graduate - master & PhD), this textbook is dedicated also to industry and research purposes. Authors
lX Faculty of Civil
Belgrade
Master Study
Faculty of Civil Engineering and Architectme
Nis Doctoral Study
COMPUTATIONAL ENGINEERING
LECTURES
LESSON I
1. Mathematics and Computer Science
1.1 Calculus
The principal topics in calculus are the real and complex number systems, the concept of limits ancl convergence, and the properties of functions. Convergence of a sequence of numbers :ri is defined as follows: The sequence :r.;, converges to the limit :c* if
7 given any tolerance E > 0, there is an index N = N(E) so that for all ·i 2:: N we have I xi-:r* I ::=;E. The notation for this is lim x.;, = x*. i--too Convergence is also a principal topics of numerical computation, but with a different emphasis.quotesdbs_dbs17.pdfusesText_23
3.3. Gauss-Seidel method . 3G
3.4. Program 37
3.5. Packages fm systems of linear algebraic equation:-; 41
Bibliography . 42
Assignment-II-III (http://www. ni. ac. yu/CDP I Assignment-II-III. pdf)4. Problems 45
4.1. Introduction 4ij
4.2. Localization of eigenvnlues 50
4.3. l'viethocls for dominant eigenvalnes 51
4.4. Niethocls for subdominant G3
4.5, problem for synnnetric tridiagonal matrices 57
4,6. LR ancl QR algorithms . :)9
4.7. Soft1va.re eigenpacka.ges GO
4.8. Generalized a.nd nonlinear eigenvalue problems G1
v 62BibliogTaphy .
Assignment-
IV5. Nonlinear Equations and Systems of Equations
5 .1. N onlineai' equations
5.1.0. Introduction
5.1.1. Newton's method
5.1.2. Bisection method
5 .1.3. Program reali,.-;ation
6565
65
68
72
73
5.2. System of nonlinear equations 83
5.2.1. Newton-Kantorowitch (R.aphson) method 83
5.2.2. Gradient method 87
5.2.3. Globally
convergent methods 91Bibliography . 94
Assignment-V (http;/ lwww. gaf. ni. ac. yuiCDP I Assignment-V. pdf)6. Approximation and Interpolation 97
6.1.Introduction 97
6.2.Chebyshev systems
6.3.Lagrange's interpolation
6.4. Newton's interpolation with divided differences
6.5. Newton's
interpolation formulas 6.6.Spline functions and interpolation by splines
6.7.Prony's interpolation
6.8. Packages for interpolation of functions
Bibliography .
Assignment-VI (http: I lwww. gaf .ni. ac. yuiCDP I Assignment-VI. pdf) 9899
100
102
104
106
107
108
7. Best Approximation of Functions 109
7.1. Introduction 109
7.2. Best L
2 approxiination7.3. Best I"· approximation
7.4. Packages for approximation of fnnctions
Bibliography .
Assignment-VII . (http: I lwww. gaf. ni. ac. yuiCDPI Assignment-VII. pdf) 112114
119
119
8. Numerical Differentiation and Integration 121
8.1. Numerical differentiation 121
8.1.1.
Introduction 121
8.1.2. Formulas for numerical differentiation 121
8.2. Numerical
integration-Quadrature formulas 12.38.2.1. Introduction 123
8.2.2.
Newton-Cotes formulas 124
8.2.3.
Generalized qHadra.tHne formulas 126
8.2.4.
integration 1288.2.5.
Program 128
8.2.6.
On numerical evaluation of a class of double integrals 1328.2.7.
Packages for nmnerical integration 134
Bil>liography . 13.5
Assignnwnt-VIII (http: I lwww. gaf. ni. ac. yuiCDP I Assignment-VIII. pdf)9. Ordinary Differential Equations -ODE 137
9.1.Introduction 1:3 7
vi0.2. Euler's method
0.3. Ge1wr;d liiwm multi-st.<p uwthod
0.4. of initial valuPs
0.:). Predict.or-corrP.ct.or llH'.tlwcls
0.6.Program n·alizat.ion of methods
0.7. RungP-Kut.t.a lll<-'t.lwds
0.8. Progrmu realization of Rnng<'-Kut.ta methods
0.0. Solution of of <-'quat.ious and equations of higher order
0.10. Bmmdary prohbus
0.11. Packag Bibliography .
Assignment-IX
10. Partial Differential Equations -PDE
10.1. Introduction
10.2. Grid method .
10.3. Laplace eqnation
10.4. Vlave eqnat.ion
10.5. Packages
for Bibliography
Assignment-X
11. Integral Equations .
11.1. Introclnction
11.2. Mc!t.lwcl of approximations
11.3. Application of q1wdra.t.me formnlas
1.38 1:30 141
1-.11 142
14.) 1;j() l;j:j 158
Hil 161
1Ci3 163
164
165
167
160
170
173
173
17:) 175
176 11.4. Program
Bibliography
Assignnwnt-XI
178
Appendices
A.l. Equations of Technical Physics
A.2. Special Functions
A.3. Numerical Methods in FEM
A.4. Numerical Methods in Informatics
vii (http://www.gaf.ni.ac.yu/CDP/ETPH.pdf) (http://www.gaf.ni.ac.yu/CDP/SPEC.pdf) (http://www.gaf.ni.ac.yu/CDP/NMFEM.pdf) (http://www.gaf.ni.ac.yu/CDP/NMINF.pdf) Preface
A course in Numerical Methods in Computational Engineering, oriented to engineering education, originates at first from the course in numerical analysis for graduate students of Faculty of Civil Engineering and Architecture of Nis (GAF), and then from course Numer ical Methods held in English language at Faculty of Civil Engineering in Belgrade in the frame of project DYNET (Dynamical Network) in common of Faculty of Civil Engineering of University of Bochum, Faculty of Civil Engineering and Architecture of University of Nis, Faculty of Civil Engineering of University Belgrade, and IZIIS (Institute for Earth quake Engineering and Seismology) of University Skopje. The subject Numerical Analysis was held in the first semester of postgraduate studies at GAF by Prof G. V. Milovanovic for years. In continuation, following Bologna process, the new structured subject entitled Numerical Analysis is be introduced to PhD students at GAF. In addition, having in n1ind that course in numerical analysis become accepted as an important ingredient in the undergraduate education in engineering and technology, it was with its main topics involved in undergraduate subject Informatics II at GAF Nis (As a collateral case, in Appendix A.4. -in electronic fonn -are given numerical methods in Informatics, what could be interesting for students of this orientation). The backbone of this script are famous books of G. V. Milovanovic, Numerical Anal ysis, Part I, II, and III, Naucna knjiga, Beograd, 1988 (Serbian). In addition, the book Programming Numerical Methods in Fortran, by G. V. Milovanovic and Dj. R. Djordjevic, University of Nis, 1981 (Serbian), with its engineering-oriented text and codes, was rather used. As previously noted, this textbook is supporting undergraduate studies, master and doctoral study at GAF; and international master study in the frame of DYNET project. Presentation on GAF site would enable distance technique and on-line consulta tions with lecturer. By up-to-day engineering oriented applications the supporting of life long education of civil engineers will be enabled. This script will be available on the site of GAF (http: I lwww. gaf. ni. ac. yu) under In ternational Projects and can be reached by chapters using address http: I lwww. gaf. ni. ac. yul cdplsubj ect_syllabus .htm. Each chapter concludes with a ba sic bibliography and suggested further reading. Tutorial exercises in form of selected as signments are also presented on the site of GAF. Some hints for solutions are given in the same files. Devoted primarily to students of Civil Engineering (undergraduate and graduate - master & PhD), this textbook is dedicated also to industry and research purposes. Authors
lX Faculty of Civil
Belgrade
Master Study
Faculty of Civil Engineering and Architectme
Nis Doctoral Study
COMPUTATIONAL ENGINEERING
LECTURES
LESSON I
1. Mathematics and Computer Science
1.1 Calculus
The principal topics in calculus are the real and complex number systems, the concept of limits ancl convergence, and the properties of functions. Convergence of a sequence of numbers :ri is defined as follows: The sequence :r.;, converges to the limit :c* if
7 given any tolerance E > 0, there is an index N = N(E) so that for all ·i 2:: N we have I xi-:r* I ::=;E. The notation for this is lim x.;, = x*. i--too Convergence is also a principal topics of numerical computation, but with a different emphasis.quotesdbs_dbs17.pdfusesText_23
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